2. STANDARD COSMOLOGY

A. The Cosmological Principle

The Cosmological Principle (CP) is the rudimentary foundation of
most standard cosmological models. The CP can be summarized by
two principles of spatial invariance. The first invariance is
isomorphism under translation and is referred to as homogeneity.
An example of homogeneity can be seen in a carton of homogeneous
milk. The milk or liquid, looks the same no matter where one is
located within it. In the realm of cosmology, this corresponds to
galaxies being uniformly distributed throughout the universe. This
uniformity would be independent of the location one chooses to
make the observations. Thus, a translation from one galaxy to
another would leave the galactic distribution invariant
(invariance under translation).

The next element of the CP is perhaps more difficult to be
realized physically. This invariance is isomorphism under
rotation and is referred to as isotropy. A simple way of
visualizing isotropy is to say that direction, such as North or
South, can not be distinguished. For example, if one were
constrained to live on the surface of a uniform sphere, there
would be no geometrical method to distinguish a direction in
space. Although, as soon as features are introduced on the sphere
(such as land masses or cracks in the surface of the sphere), the
symmetry is lost and direction can be established. This fact gives
a clue that isotropy, as you might have guessed, is closely
related to homogeneity.

The concepts of homogeneity and isotropy may appear contradictory
to local observation. The Earth and the solar system are not
homogeneous nor isotropic. Matter clumps together to form objects
like galaxies, stars, and planets with voids of near-vacuum in
between. However, when one views the universe on a large scale,
galaxies appear `smeared out' and the CP holds.

Experimental proof of isotropy and homogeneity has been approached
using a number of methods. One of the most convincing
observations is referred to as the Cosmic Background Radiation
(CBR). In the standard Big Bang model, the universe began at a
singularity of infinite density and infinite temperature. As the
universe expanded it began to cool allowing nucleons to combine
and then atoms to form. About 300,000 years after the Big Bang,
radiation decoupled from matter, allowing it to `escape' at the
speed of light. This radiation continues to cool to the present
day and is observed as the CBR. Any observation of the CBR gives
a picture of the mass distribution at around 300,000 years. The
temperature of the CBR, first predicted theoretically in the
1960's by Alpher and Herman at 5K
[1], and Gamow at a
higher 50K [2],
was not taken seriously. A later prediction by Dicke, et. al.
[3] yielded
3 K,
but as Dicke and colleagues set out to measure this remnant
radiation, they found someone had already made this measurement.
Dicke remarked, ``Well boys, we've been scooped''
[4].

The first successful measurement of the CBR was made in 1964 by
Penzias and Wilson, two scientists working on a satellite
development project for Bell Labs
[4]. Their
measurements revealed that the CBR was characteristic of a
black-body with a corresponding temperature of around 3K as
illustrated in Figure (1)
[4]. The
measured wavelengths were on the order of 7.35 cm, corresponding
to the microwave range of the electromagnetic spectrum. The CBR in
this range is referred to as the Cosmic Microwave Background
(CMB)(1)

Figure 1. The cosmic background spectrum is
that of a near perfect
blackbody as predicted by theory. The above graph represents the
most recent observations by a number of collaborations. Graph
courtesy of Wayne Hu
[64].

Another important observation of Penzias and Wilson is the fact
that the CMB is uniform (homogeneous) in all directions
(isotropic). Thus, the CMB offers an experimental proof of the
isotropy and homogeneity of the universe.

Because of its importance, further measurements of the CBR have
been carried out. One such project named COBE, for Cosmic
Background Explorer, in 1989, measured the CBR to have a
temperature of 2.73 K and a distribution that is isotropic to
one part in 105
[5]. COBE also
has the distinction
of being the first satellite dedicated solely to cosmology. Future
measurements will be made by dedicated satellites like COBE, but
these satellites will have much higher angular resolution. They
are planned to be launched around the beginning of the century.
One program directed by NASA is called MAP, for Microwave
Anisotropy Probe(2)
and another named the PLANCK Explorer
is being launched by the European Space
Agency(3)
The accurate measurement of the CBR offers an observational test
of cosmological models, as well as, the CP.

In addition to these benefits of CBR observations, the CBR can
also be used to setup a Cosmic Rest Frame (CRF). This concept is
reminiscent to the ideas of Ernst Mach. One chooses a reference
frame to coincide with the Hubble expansion, i.e., with the motion
of the average distribution of matter in the universe. It is
convenient to define our coordinates in this frame to save
confusion in measurements such as the expansion of spacetime and
the Hubble Constant; however, these coordinates are in no way
`absolute' coordinates. Using the CBR to define the CRF and taking
galaxies as the test particles of the model serves to greatly
simplify the dynamics in an expanding universe. The CRF is used to
ease calculations and make the interpretation of the dynamics of
an expanding universe more tractable.

The current and proposed measurements of the CBR offer a
convincing test of the homogeneity of space. Measurements of the
temperature of the CBR are uniform to one part in 105. This
suggests the universe is homogeneous and isotropic to a high
degree of accuracy. However, since this measurement is taken from
our (the Earth's) vantage point, one can not assume the same
conclusion from another vantage point. This can be remedied by
considering how the CBR is related to the distribution of matter
at the time the photons of the CBR decoupled. This offers a `snap
shot' of the inhomogeneities in the density of the universe. If
these regions contained more inhomogeneity, galaxies would not be
visible today. This idea will be discussed in more detail later;
as an alternative one can introduce the Copernican Principle (CP).

The CP states that no observers occupy a special place in the
universe. This appears to be a favorable prediction, based on the
evidence above, as well as lessons coming from the past. For
example, the correct model of the solar system was not realized
until humans realized they were not the center of the solar
system. This may be a bit humbling to the human ego, but the
Copernican Principle, along with homogeneity and isotropy, serve
to greatly simplify the number of possible cosmological models for
the universe. Later, it will be seen that homogeneity follows
naturally from inflation. If the universe went through a brief
period of rapid expansion, the fact that galaxies exist at all
will be a necessary and sufficient condition for a homogeneous universe.

There is also the proposal for cosmic `no-hair' theorems. These
theorems are similar to the `no-hair' proposal of black holes,
which predict that any object that contains an event horizon will
yield a Schwartzschild spherically symmetric solution at the
singularity. The Big-Bang singularity is no exception, and the
event horizon is the Hubble distance to be explored in sections to
come. For now, experiment suggests that it is safe to assume the
Copernican Principle is valid.

Below is a brief descriptive summary of observational methods for
testing the CP:

Particle Backgrounds - These observations represent the
strongest argument for isotropy and homogeneity. As the universe
evolved it cooled allowing various particle species to become
`frozen out', meaning that the particles were freed from
interactions. Photons, for example, became frozen out at the time
of decoupling and are visible today as the CBR. These backgrounds
serve as an important experimental test for predictions by various
cosmological models.

The Observed Hubble Law - This law states that the farther
away a galaxy
is, the faster it will be observed to recede. This phenomena is
observed through a redshift which will be described in a later
section. The observed redshift, first witnessed by Edwin Hubble
was the first indication that the universe obeys the CP.

Source Number Counts - Of all methods this is the most
uncertain at this time. This method requires collecting light
from galaxies and inferring whether `clustering' occurs. One
debate over the accuracy of such methods is based on the idea that
most matter in the universe might be of a non-luminous type, the
so-called Dark Matter. Another problem is that current technology
does not allow observations at distances far enough to get a good
sample of the population. However, this technique shows promise
for the future, and the SLOAN(4)
Digital Sky Survey is a current project that will map in detail
one-quarter of the entire sky, determining the positions and
absolute brightness of more than 100 million celestial objects.
It will also measure the distances to more than a million galaxies
and quasars.

Inflation - Although it is premature at this point to
discuss observational consequences of inflation, it will be shown
that inflation predicts small perturbations in the universe that
result in the large-scale structure observed today. It will be
shown that if these perturbations were too large then the
structure we observe today would not be possible. Thus, if
inflation can be proved through observation, it would imply the
universe must have been very homogeneous at the time of
decoupling.

The established concepts of the CP aid in simplification of
cosmological models, but a further simplification can be made by
invoking the Perfect Cosmological Principle. This principle
differs from the previous one in that it assumes temporal
homogeneity and isotropy. This would imply a static universe, for
if the universe were expanding or contracting it would not look
the same now, as it did in the past. However, one exception that
will prove important later is the case of a (anti or quasi)
DeSitter Space. By the observations of Edwin Hubble and the
theoretical work by Lamaître(5)
it was
shown that the expansion of the universe is an accurate
assumption. CP models further suggest that a static universe
would be as stable as a pencil standing on its end. Thus, the
Perfect Cosmological Principle does not appear to be an acceptable
assumption within the standard model
[7].
(6)

The last element to be discussed concerning the CP is the Weyl
Postulate. This postulate formally states that, ``the world lines
of galaxies designated as `test particles' form a 3-bundle of
nonintersecting geodesics orthogonal to a series of spacelike
hypersurfaces''
[8]. In other
words, the geodesics
on which galaxies travel do not intersect. This adds another
symmetry to the picture of the expanding universe allowing
simplification of the spacetime metric and the Einstein equations.

1 The significance in making this
distinction will
manifest itself later, but it is worth noting that other
backgrounds are measurable and offer further evidence of the CP.
Back.
2 For more info see:
http://map.gsfc.nasa.gov/. Back.
3http://astro.estec.esa.nl/SA-general/Projects/Planck/Back.
4http://www.sdss.orgBack.
5 Lamaître will not be
mentioned further but it is worth noting that his work and
persistence, backed by the experimental efforts of Hubble, were
instrumental in convincing Einstein that the universe was indeed
expanding. After this persuasion, Einstein was quoted as saying
this was the biggest mistake of his career
[6].
Back.
6 This
is not totally correct. In some space-times, such as anti-DeSitter
space, there exists temporal homogeneity. For a rigorous
treatment of such space-times consult
[7].
Back.